Solving For Y In The Equation -3/(2y-12) - 1 = -5/(y-6)

This article provides a step-by-step guide to solving the algebraic equation $-\frac{3}{2y-12} - 1 = -\frac{5}{y-6}$ for the variable y. We will explore the methods used to isolate y and find the solution while discussing potential pitfalls and the logic behind each step. This comprehensive explanation aims to help students and anyone interested in algebra to understand how to approach and solve similar equations effectively.

1. Initial Equation and Simplification

Our primary goal here is to solve the given equation:

$-\frac{3}{2y-12} - 1 = -\frac{5}{y-6}$.

Before diving into the algebraic manipulation, let's first identify the key components and potential issues. We have fractions with denominators involving y, specifically $2y - 12$ and $y - 6$. Notice that $2y - 12$ can be factored as $2(y - 6)$. This common factor hints at a possible simplification strategy that we will leverage. Factoring this common factor is crucial for simplifying and solving the equation.

The initial step involves simplifying the equation by addressing the fractions and combining like terms. Recognizing the common factor allows us to streamline the process. By factoring out the 2 from $2y - 12$, we get $2(y - 6)$. This observation will be critical when we try to eliminate the denominators later on. The given equation is now seen in a more digestible form, ready for the next steps in solving for y. Addressing the fractional terms and potential simplifications early on is vital for a clearer path towards the solution. The factored form of the denominator will greatly aid in finding a common denominator and simplifying the equation. So, before we proceed further, let us take note of this simplified form.

2. Finding the Common Denominator and Eliminating Fractions

The next crucial step is to eliminate the fractions in the equation. To do this, we need to find a common denominator for all terms. Recall the equation:

$-\frac{3}{2(y-6)} - 1 = -\frac{5}{y-6}$.

As we noticed earlier, the denominators are $2(y - 6)$ and $(y - 6)$. The least common denominator (LCD) is thus $2(y - 6)$. Now, we'll multiply every term in the equation by this LCD. This process will clear the fractions, making the equation easier to work with.

When we multiply each term by $2(y - 6)$, the fractions will cancel out, leaving us with a simpler equation involving only the numerators and the LCD. This step is pivotal in transforming the equation into a more manageable form. By multiplying both sides of the equation by the LCD, we ensure that the equality is maintained while simultaneously eliminating the fractions. This is a common and effective strategy for solving equations involving rational expressions. After multiplying each term by the LCD, we should have an equation that is free of fractions, making it significantly easier to solve for y.

Multiplying the first term $-\frac{3}{2(y-6)}$ by $2(y - 6)$ cancels the denominator, leaving us with $-3$. Multiplying the second term, $-1$, by $2(y - 6)$ gives us $-2(y - 6)$. Lastly, multiplying the third term $-\frac{5}{y-6}$ by $2(y - 6)$ cancels the $(y - 6)$ term, leaving us with $-5 \cdot 2 = -10$. So, the transformed equation, free from fractions, will be a stepping stone towards finding the value of y.

3. Simplifying the Equation

Having eliminated the fractions, our equation now looks like this:

$-3 - 2(y - 6) = -10$.

The next step is to simplify this equation by distributing and combining like terms. The goal here is to get the equation into a form that is easier to solve for y. We'll start by distributing the $-2$ across the parentheses $(y - 6)$. This will remove the parentheses and allow us to combine the constant terms.

Distributing the $-2$ in the term $-2(y - 6)$ means multiplying $-2$ by both y and $-6$. This gives us $-2y + 12$. Replacing the term $-2(y - 6)$ with $-2y + 12$ in the equation, we get $-3 - 2y + 12 = -10$. This simplification step is crucial for isolating y and finding its value. Now, we have an equation where we can combine like terms, specifically the constants, to further simplify it. Combining like terms will help us streamline the equation and bring it closer to the standard form for solving linear equations.

Combining the constant terms $-3$ and $+12$ on the left side of the equation, we have $-3 + 12 = 9$. So, the equation becomes $9 - 2y = -10$. This simplified form is much easier to work with, and we are one step closer to isolating y. The simplification process ensures that we are dealing with the most basic form of the equation before we proceed with solving for the variable. With the constants combined, we can now focus on isolating the term containing y.

4. Isolating the Variable

Now that we have the simplified equation $9 - 2y = -10$, we need to isolate the term containing y. The goal is to get the term with y alone on one side of the equation. We can do this by subtracting 9 from both sides of the equation. This operation will maintain the equality and move the constant term to the right side of the equation.

Subtracting 9 from both sides of the equation $9 - 2y = -10$ will cancel out the +9 on the left side, leaving us with $-2y$ on the left. On the right side, we have $-10 - 9$, which equals $-19$. So, the equation becomes $-2y = -19$. This step is crucial for isolating y and bringing us closer to the solution. By isolating the term with y, we simplify the process of solving for y itself. Now, we only have one step left to find the value of y.

Having isolated the term with y, $-2y = -19$, we are now ready to solve for y. This involves one final step: dividing both sides of the equation by the coefficient of y, which is $-2$. This division will isolate y on the left side and give us its value on the right side. The isolation of the variable is a fundamental step in solving any algebraic equation.

5. Solving for y

With the equation $-2y = -19$, the final step is to solve for y. To do this, we divide both sides of the equation by $-2$:

$\frac{-2y}{-2} = \frac{-19}{-2}$.

Dividing both sides by $-2$ will isolate y on the left side. On the right side, dividing $-19$ by $-2$ results in a positive value since a negative divided by a negative is positive. This division is the final operation needed to find the value of y. By performing this division, we determine the solution to the equation, thereby fulfilling our primary goal.

Performing the division, we find that $y = \frac{19}{2}$. This is the solution to the equation. We have successfully isolated y and found its value. It's always a good practice to check our solution by substituting it back into the original equation to ensure it holds true. This verification step confirms the accuracy of our solution and helps avoid any errors that may have occurred during the solving process.

6. Checking the Solution

To ensure the solution $y = \frac{19}{2}$ is correct, we substitute it back into the original equation:

$-\frac{3}{2y-12} - 1 = -\frac{5}{y-6}$.

Substituting $y = \frac{19}{2}$ into the equation, we get:

$-\frac{3}{2(\frac{19}{2})-12} - 1 = -\frac{5}{\frac{19}{2}-6}$.

Now, we simplify each side of the equation to verify that they are equal. This verification step is crucial to confirm that our solution is accurate and that no errors were made during the solving process. Substituting the solution back into the original equation helps us catch any potential mistakes in our calculations.

First, let's simplify the left side of the equation. We have $2(\frac{19}{2}) = 19$, so the denominator becomes $19 - 12 = 7$. Thus, the left side is $-\frac{3}{7} - 1$. To combine these terms, we need a common denominator, which is 7. So, we rewrite 1 as $\frac{7}{7}$, giving us $-\frac{3}{7} - \frac{7}{7} = -\frac{10}{7}$.

Next, let's simplify the right side of the equation. We have $\frac{19}{2} - 6$. To subtract these, we need a common denominator, which is 2. So, we rewrite 6 as $\frac{12}{2}$, giving us $\frac{19}{2} - \frac{12}{2} = \frac{7}{2}$. Thus, the right side is $-\frac{5}{\frac{7}{2}}$. To divide by a fraction, we multiply by its reciprocal, so we have $-5 \cdot \frac{2}{7} = -\frac{10}{7}$.

Since both sides simplify to $-\frac{10}{7}$, the solution $y = \frac{19}{2}$ is correct. This verification confirms that we have accurately solved the equation and that our solution satisfies the original problem statement. The process of checking the solution is an essential part of problem-solving in algebra and other areas of mathematics.

7. Conclusion

In summary, we have successfully solved the equation $-\frac{3}{2y-12} - 1 = -\frac{5}{y-6}$ for y. The steps involved simplifying the equation by finding a common denominator, eliminating fractions, combining like terms, isolating the variable, and finally, solving for y. We also verified our solution by substituting it back into the original equation.

The process of solving algebraic equations like this is a fundamental skill in mathematics. It involves a systematic approach, careful execution of algebraic manipulations, and a thorough understanding of the underlying principles. The solution we found is $y = \frac{19}{2}$, which satisfies the given equation. The ability to solve such equations is crucial for more advanced topics in mathematics and its applications in various fields.

This comprehensive guide aimed to provide a clear and detailed explanation of each step involved in solving the equation. By understanding these steps, students and anyone interested in algebra can confidently tackle similar problems and enhance their problem-solving skills. The combination of algebraic manipulation and logical reasoning is key to success in mathematics, and this guide has demonstrated how these skills can be applied effectively to solve algebraic equations.